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Model for classical and ultimate regimes of radiatively driven turbulent convection.
Mathieu Creyssels
To cite this version:
Mathieu Creyssels. Model for classical and ultimate regimes of radiatively driven turbulent convection..
2020. �hal-02299927v2�
Model for classical and ultimate regimes of radiatively driven turbulent convection
M. Creyssels †
Laboratoire de M´ecanique des Fluides et d’Acoustique, Ecole Centrale de Lyon, Univ. Lyon, CNRS, 69134 Ecully, France
(Received xx; revised xx; accepted xx)
In a standard Rayleigh-B´enard experiment, a layer of fluid is confined between two horizontal plates and the convection regime is controlled by the temperature difference between the hot lower plate and the cold upper plate. The effect of direct heat injection into the fluid layer itself, for example by light absorption, is studied here theoretically. In this case, the Nusselt number (N u) depends on three non-dimensional parameters: the Rayleigh (Ra) and Prandtl (P r) numbers and the ratio between the spatial extension of the heat source (l) and the height of the fluid layer (h). For both the well known classical and ultimate convection regimes, the theory developed here gives a formula for the variations of the Nusselt number as a function of these parameters. For the classical convection regime, by increasing l/h from 0 to 1/2, N u gradually changes from the standard scalingN u∼Ra1/3to an asymptotic scalingN u∼Raθ, withθ= 2/3 orθ= 1 by adopting, respectively, the Malkus (1954) theory or the Grossmann & Lohse (2000) theory. For the ultimate convection regime, N u gradually changes from N u ∼ Ra1/2 scaling to an asymptotic behaviour seen only at very highRafor whichN u∼Ra2. This theory is validated by the recent experimental results given by Bouillautet al.(2019) and also shows that for these experiments, Ra and Re numbers were too small to observe the ultimate regime. The predictions for the ultimate regime cannot be confirmed at this time due to the absence of experimental or numerical work on convection driven by internal sources and for very largeRa numbers.
1. Introduction
Rayleigh-B´enard (RB) convection is a classical fluid dynamics problem and has been the subject of numerous experimental, theoretical and numerical studies. When Rayleigh numbers are high (generally above 106), two distinct theories, called classical and ulti- mate, give two distinct asymptotic behaviours for the Nusselt number as a function of the Rayleigh number. The classical theory states that the heat flux should be independent of the height of the fluid layer leading from the definition ofN uandRato the following asymptotic law: N u∼Ra1/3(Malkus 1954; Priestley 1954; Grossmann & Lohse 2000).
The ultimate theory asserts that for very high Rayleigh numbers, the heat flux should become independent of the fluid dissipative coefficients ν and κ giving an asymptotic law likeN u∼Ra1/2 (Kraichnan 1962; Spiegel 1971; Siggia 1994; Chavanneet al. 1997;
Ahlerset al.2009; Grossmann & Lohse 2011; Chill`a & Schumacher 2012).
This paper is an extension of RB theories in the case of a heat source spatially distributed within the fluid layer. An example of this kind of heating is given by Lepot et al. (2018); Bouillautet al. (2019). The authors experimentally developed a new RB cell concept for which heat is not injected through thermal conduction between the lower
† Email address for correspondence: [email protected]
l
l
δT
δT
l
l
δT
δT
qv(z) Φ(z)
T(z)
qv(z) Φ(z)
T(z)
z z z z z z
−Q/l
0 Q/l 0 Q 0 Q/l 0 Q
−Q/l Tc
Th Th
Tc
Tb Tb
Figure 1.Modified RB experiment in the case ofl/δT <1 (left) and in the case of l/δT >1 (right). The heat is injected in volume near the lower plate (red zone) while the fluid is cooled in volume near the upper plate (blue area), both with a characteristic lengthl. The two thermal boundary layers with a width of δT are also displayed (hatched areas). The profiles of the volumetric (positive and negative) power source (qv), the mean heat flux (Φ) and the mean temperature (T) are also shown for each case.
heating plate and the fluid above it. In their experiment, the lower plate is transparent and the working fluid is a homogeneous mixture of water and dye. A powerful spotlight placed under the lower plate shines through the fluid, and the light, after passing through the transparent plate, is absorbed by dye and therefore by the fluid located near the plate.
According to the Beer-Lambert law, this kind of heating corresponds to a volume heat source that decays exponentially from the lower plate to a characteristic heightl, leading to a local heating of the following form:
qv(z) = Q l exp
−z l
, (1.1)
where Qis the total heat flux radiated by the spotlight into the fluid (in W/m2) andz is the vertical coordinate withz= 0 on the lower plate. The characteristic heightl can be changed since it is inversely proportional to the dye concentration. Hereafter, (1.1) is assumed to be valid even if the model proposed in this article can easily be generalized to other forms of local heating rates.
Lepotet al.(2018); Bouillautet al.(2019) and Doering (2019) claimed that the study of this type of modified RB experiments should allow progress in understanding turbulent convection in both natural flows and a conventional RB cell. Indeed, in many geophysical and astrophysical flows, convection is driven by internal heating due to, for example, the radioactive decay in the Earth’s mantle or the thermonuclear reactions in stars. It is therefore easy to understand that a modified RB experiment is a first approach to model turbulent flows in natural systems even ifRanumbers are very different. In addition this work also aims to provide interesting information on turbulent convection. Indeed, heat transport in a conventional RB cell is essentially controlled by the thermal boundary layers near the plates and their stability explains the difference between the two theories of convection (the classical and the ultimate). To investigate these boundary layers, the location of the heat sources can be easily changed by adjusting the absorption height l (Lepotet al. 2018; Bouillaut et al. 2019). This is a similar approach to that used by other authors, which consists of replacing the lower and upper plates with rough plates (Shen et al. 1996; Roche et al. 2001; Qiu et al. 2005; Stringano et al. 2006; Tisserand
et al. 2011; Zhu et al. 2017, 2019). Roche et al. (2001) and Tisserand et al. (2011) reported an increase of the N u vs Ra scaling exponent from 1/3 to 1/2, even if the range ofRaexplored and their interpretation of it was very different. Rocheet al.(2001) interpreted the transition for the exponent to the value 1/2 as a turbulent transition for the thermal boundary layers because the Ra numbers were high (>1012) and the transition was already observed with smooth plates. On the contrary, Tisserand et al.
(2011) interpreted the increase in the exponent as a destabilization by buoyancy of the fluid placed between the rough elements. The observation of the exponent 1/2 is then fortuitous in the latter case and, as underlined by Zhu et al. (2017); Rusaou¨en et al.
(2018), the exponent 1/2 can only be seen over a limited range of the Rayleigh number.
By increasingRa further, the exponent decreases and returns to its classical value close to 1/3. Note that the range ofRa for which exponent 1/2 is observed can be increased using multi-scale roughness (Zhuet al.2019).
In this theoretical study, a model is proposed to deduce scaling laws of the Nusselt number as a function of the three non-dimensional parameters that control turbulent convection i.e. Ra, P r and the ratio of absorption height to cell height (˜l = l/h). In a standard RB experiment, both plates play the same role (for a small temperature difference and by adopting the Boussinesq approximation) and the corresponding thermal boundary layers have the same behaviour and therefore the same width (δT). To have two similar boundary layers in a modified RB cell, the upper part of the cell must be cooled with the same power profile as that used for the heating process, so qv(z) =
−Ql exp(−h−lz). The injected or extracted power profile is shown in Fig. 1 for both cases l/δT < 1 (left) and l/δT > 1 (right). When l → 0, this experiment becomes a standard RB experiment while, when the lengthlincreases, the lower and upper thermal boundary layers are heated and cooled respectively. Finally, when l becomes greater thanδT, the bulk flow is also heated and cooled simultaneously since the lower region is heated while the upper region is cooled (Fig. 1 right). The Rayleigh number in a modified RB experiment can be defined as in a conventional RB cell by using the temperature difference between the two plates (∆T = Th−Tc), between the lower plate and the mean bulk flow (∆T = 2(Th−Tb)) or between the mean bulk flow and the upper plate (∆T = 2(Tb−Tc)). When Rayleigh numbers are high, it is assumed that the convective flow of a modified RB experiment is strong enough to impose an almost constant mean temperature over time in the bulk flow,i.e. outside the boundary layers (see Fig. 1), as experimentally observed in a standard RB experiment.
A major difference between modified and standard RB experiments concerns the mean heat flux through the cell from the bottom plate to the top plate. Indeed, when a steady state is reached, the heat flux averaged over a horizontal section must be independent of the vertical coordinate (z) for a standard RB experiment, whereas for a modified RB cell, this heat flux cannot be constant even in a steady state. When considering a horizontal slice of fluid, the energy given in volume must be evacuated outside the slice, which requires a gradient of the mean heat flux in the fluid (see Fig. 1). For z = 0 and z = h, the heat flux is zero because the two horizontal plates are assumed to be perfectly insulated. Far from the plates, in the center of the cell wherel ≪z ≪h−l, the volumetric heat sourceqv is close to 0, and energy conservation leads to a heat flux equal to Q~ez. Thus, with the exception of the blue and red regions shown in Fig. 1,Q represents the heat flux through the cell and the Nusselt number can be defined as in a standard RB experiment as
N u= Qh
λ∆T, (1.2)
whereλis the thermal conductivity of the fluid andhthe height of the cell. As previously mentioned, when ˜l →0,N utends towards the Nusselt number that can be obtained in the same cell but with standard RB conditions that are a constant heat flux and fixed temperatures at both plates. Hereafter, this Nusselt number will be taken as a reference and calledN u0(Ra, P r) = lim˜l→0N u(Ra, P r,˜l).
Finally, it is questionable whether this type of modified RB experiment can be per- formed experimentally. Indeed, heating in volume can be achieved using either strong light (Lepotet al. (2018)), an electric current or even by fixing heating elements in the fluid (Kulacki & Goldstein (1972); Goluskin (2015); Goluskin & van der Poel (2016)).
On the contrary, cooling in volume is more difficult to achieve experimentally. However, Lepot et al. (2018); Bouillaut et al. (2019) have shown that, in their experiments, turbulent convection develops quasi-stationary internal temperature gradients leading to a temperature difference between the lower plate and the bulk flow that is almost constant over time (see Fig. 1 B in Lepotet al.(2018)). Therefore, the theoretical results given below will be compared in section 4 with those obtained experimentally by Lepot et al. (2018); Bouillautet al. (2019). The theoretical model is based, on the one hand, on the known structure of the flow and temperature fields observed experimentally and numerically in a standard RB cell at high Rayleigh numbers (generally>106), on the other hand, on the different theories of RB convection given in the literature.
2. Background on N u vs Ra scalings for standard RB convection
For high Rayleigh numbers, convective flow is turbulent almost everywhere in the cell except in two thin thermal boundary layers located against the lower and upper plates.This dynamic structure of the flow yields to a particular field for the mean temperature.
Indeed, in the bulk flow, turbulent convection produces large temporal and spatial variations for temperature fluctuations but an almost uniform mean temperature field withTb= (Th+Tc)/2 for symmetry reasons and assuming the Boussinesq approximation is valid (the mean temperature profile is represented in Fig. 1). On the contrary, the mean temperature increases or decreases by∆T /2 = (Th−Tc)/2 in each boundary layer.
Therefore, the heat transfer averaged over a horizontal section is dominated by turbulent convection in the bulk flow (Φ ≈ρcpw′T′, where w′ and T′ are the fluctuations of the vertical velocity and temperature respectively), whereas the heat transfer is driven by thermal conduction in the two thin boundary layers (Φ≈ −λ∂T /∂z, where T(z) is the temperature averaged both on time and on a horizontal section located at the distancez from the plate). The thickness of each thermal boundary layer (δT) is controlled by the temperature difference ∆T /2 and the mean heat flux assuming that Φ can be written as Φ = λ∆T /(2δT). This last equation is valid regardless of the convection regime or the adopted theory (see Kraichnan (1962) and Grossmann & Lohse (2000)), leading to a ratioδT/hdepending only on the Rayleigh number as
δT
h = 1
2N u0(Ra, P r). (2.1)
2.1. Classical regime by Malkus (1954)
The first regime of convection, called classical, was proposed by Malkus (1954);
Priestley (1954). It has the merit of simplicity and predicts a scaling lawN u0∼Ra1/3, hence with an exponent 1/3 close to the exponents observed both in the experiments and the numerical simulations in the range ofRa between 106and 1012. This regime of convection is entirely characterized by a constant Rayleigh number for each boundary
layer as:
gα∆T δ3T
2νκ =Ra∗. (2.2)
Using (2.1) and (2.2), we obtain for the classical regime:
N u0= Ra
24Ra∗ 1/3
. (2.3)
2.2. Ultimate regime by Kraichnan (1962)
For very large Rayleigh numbers, the thermal boundary layers observed in the case of the classical regime can be destabilized and Kraichnan (1962); Spiegel (1971) assumed that they could become similar to the velocity boundary layers observed in the case of a fully developed mean shear flow. This ultimate regime is then characterized by a constant but Prandtl-dependent P´eclet number for each thermal boundary layer:
v∗0δT
κ =P e∗(P r), (2.4)
where κ = λ/(ρcp) is the thermal diffusivity of the fluid. For small Prandtl numbers, the thickness of the viscous sublayer is smaller than δT leading to a constant P´eclet number P e∗ =P e∗P r→0. On the contrary, at moderateP r numbers,P e∗ varies as√
P r sinceP e∗=p
P e∗P r→0ResP r, whereResis the characteristic Reynolds number for the top of the viscous sublayer (Kraichnan 1962). The new unknown parameter v∗0 can be interpreted as a friction velocity and measures the rms value of velocity fluctuations at the edge of each boundary layer, similarly to the friction velocity defined in the case of a channel flow. Unlike the classical regime for which the characteristic Rayleigh number Ra∗ depends only on ∆T and δT, P e∗ is linked to the convective flow in the bulk by the velocity fluctuations v∗0. Thus, determining the Nusselt number for the ultimate regime requires additional assumptions and equations. The parameterv0∗is an increasing function of the large-scale mean velocity (U0), also called as the wind turbulence. By analogy with what is well known for the channel flow, Kraichnan (1962) assumed that v0∗∼U0/lnRe0, withRe0=U0h/ν. In addition, the wind velocity is obtained by writing that the Richardson number in the bulk flow is of order 1,i.e. Ri=gα(w′T′)h/U03∼1.
Using the definitions ofRe0,Ra andN u0, this last equation yields to Re30∼RaN u0
P r2 . (2.5)
We can note that (2.5) is valid both for the classical regime obtained by Malkus (1954), the ultimate regime proposed by Kraichnan (1962), and the two convection regimes (II and IV) of the Grossmann & Lohse (2000) theory (see section 2.3). Using (2.1), (2.5) and v0∗∼U0/lnRe0, (2.4) becomes
Re20ln(Re0)∼ Ra
P rP e∗. (2.6)
Then, using (2.6), (2.5) gives the Nusselt number for the ultimate regime:
N u0∼
P r Ra [P e∗ln(Re0)]3
1/2
. (2.7)
For small P r numbers (typically P r < P e∗P r→0/Res), N u0 ∼ P r1/2Ra1/2/(lnRe0)3/2 while for moderateP r numbers,N u0∼P r−1/4Ra1/2/[ln(Re0)]3/2.
2.3. RB theory by Grossmann & Lohse (2000)
Grossmann & Lohse (2000) (GL) proposed a RB theory to describe with more precision the Rayleigh and Prandtl dependence of the Nusselt number. The kinetic energy and thermal dissipation rates which are defined respectively asǫu= ν2P
i,j(∂jui+∂iuj)2and ǫT =κP
i(∂iT)2play a central role in GL theory. In steady state and averaging over the whole RB cell the two equations of conservation of the turbulent kinetic energy (12P
iu2i) and of the square of the temperature give the following two exact relations:
hǫui= ν3 h4
(N u0−1)Ra
P r2 , (2.8)
hǫTi=κ ∆T
h 2
N u0. (2.9)
The key idea of the GL theory is to split both mean dissipation rates into two contribu- tions each, one from the bulk (Bu) and one from the boundary layers (BLs) as
hǫui=hǫuiBu+hǫuiBL, (2.10) hǫTi=hǫTiBu+hǫTiBL, (2.11) where
hǫuiBu= 1 h
Z h−δu
δu
ǫu(z)dz and hǫTiBu= 1 h
Z h−δT
δT
ǫT(z)dz (2.12) are, respectively, the viscous and thermal dissipation taking place in the bulk flow.
Whereas the viscous and thermal dissipation taking place in the boundary layers can be written as:
hǫuiBL= 2 h
Z δu
0
ǫu(z)dz and hǫTiBL= 2 h
Z δT
0
ǫT(z)dz. (2.13) In (2.12) and (2.13), the kinetic energy and thermal dissipation rates are first averaged over a horizontal cross-section givingǫuandǫT, respectively. The thickness of the thermal BLs (δT) is given by (2.1) while a Blasius-type layer is assumed for the viscous BLs, with a thickness of
δu
h = a
√Re0
. (2.14)
Note that the prefactor a is obtained by match with a record of experimental results (Stevenset al.2013).
To obtain the Rayleigh dependence of the Nusselt and Reynolds numbers, ǫu andǫT
need to be estimated both in the bulk flow and in the BLs:
hǫuiBu∼ U02 h/U0
1−δu
h
≈ ν3
h4Re30, (2.15)
hǫTiBu∼ (∆T)2 h/U0edge
1−δT
h
≈κ ∆T
h 2
Re0P rf
2aN u0
√Re0
, (2.16)
hǫuiBL∼ν U0
δu
2
δu
h = ν3 h4
Re5/20
2a , (2.17)
hǫTiBL∼κ ∆T
δT
2
δT
h = 2κ ∆T
h 2
N u0. (2.18)
In (2.16), the relevant velocity at the edge between thermal BL and the thermal bulk can be less thanU0, depending on the ratio:δu/δT = 2aN u0/√
Re0. Grossmann & Lohse
(2001) introduced a function 06f 61 saying that the relevant velocity at the edge then becomes U0edge =U0f(δu/δT), with f → 1 when δu/δT →0 and f → 0 forδu ≫ δT. They gavef(x) = (1 +xn)−1/n, withn= 4 as an example of functionf.
Grossmann & Lohse (2001) have also extended those estimations of the viscous and dissipation rates for very large Prandtl numbers for which (2.14) cannot stay valid.
Indeed, whenP ris high enough,δumust saturate to a maximum valueδu(Rec) lower than the height of the cell. The critical Reynolds numberRecwas estimated from experimental data to 0.28 by Grossmann & Lohse (2001) and 0.35 by Stevenset al. (2013). However, for the sake of simplicity, only the case ofRe0≫Rec is considered here.
From decomposition of the two global dissipation rates (2.10) and (2.11), four regimes of convection can be defined depending on whether the bulk or the BL contributions dominate the global dissipations. Besides, each of these four regimes is in principle divided into two subregimes, depending on whether the thermal BL or the kinetic BL is larger.
The two hǫuiBu bulk-dominated regimes (referred to as II and IV) are first presented because most of the experimental and numerical results fall under one of these two regimes (see figure 8 from Stevenset al.(2013)).
2.3.1. Regimes II and IV,hǫui ∼ hǫuiBu
For regimes II and IV, the kinetic energy dissipation rate is dominated by its bulk contribution. Combining (2.8) and (2.15), and assuming N u0 ≫ 1, we obtain (2.5) again. Regime IV is obtained for highRanumbers for which thermal dissipation rate is dominated by its bulk contribution. Combining (2.9) and (2.16), it yields to:
N u0∼Re0P rf
2aN u0
√Re0
(RegimeIV). (2.19)
For lowerRanumbers, the thermal dissipation rate is dominated by its BL contribution.
However, combining (2.9) and (2.18) yields to a trivial equation for N u0. To obtain a scaling relation betweenN u0andRe0, Grossmann & Lohse (2000) proposed to consider, in each thermal BL, the order of magnitude of the different terms of energy equation:
ux∂xT+uz∂z=κ∂zzT. (2.20) Both terms on the left-hand side are of order U0edge∆T /h whereasκ∂zzT ∼ κ∆T /δ2T. Hence, using (2.1), one gets
N u0∼ s
Re0P rf
2aN u0
√Re0
(RegimeII). (2.21)
Combining (2.5) and (2.19) or else (2.5) and (2.21), we obtain:
N uθ0i∼(N u0RaP r)1/3f
2a(N u0RaP r)1/3 (Ra/N u0)1/2
, withθII = 2 andθIV = 1. (2.22) For Prandtl numbers small or large enough,f(x)≈1 (δT ≫δu) orf(x)≈1/x(δu≫δT), and (2.22) can be simplified as follows:
N u0∼
( (RaP r)1/(3θi−1), forδT ≫δu, (2.23a) Ra1/(2θi+1), forδu≫δT. (2.23b) We can note that the sub-regimeIVu (δu ≫δT) gives the same scaling as predicted by Malkus (1954), i.eN u0∼P r0Ra1/3.
2.3.2. Regimes I and III, hǫui ∼ hǫuiBL
For these two regimes, (2.5) needs to be replaced by Re5/20 ∼RaN u0
P r2 . (2.24)
Equation (2.24) is obtained by combining (2.8) and (2.17). As for regimeIV, thermal dissipation rate is dominated by its bulk contribution in regimeIII and (2.19) is valid.
On the contrary, in regimeI, we use (2.21) instead of (2.19), as for regimeII. The Ra- andP r-dependent Nusselt number is then given by
N uθ0i∼(N u0Ra√
P r)2/5f
(2a(N u0Ra√ P r)2/5 [Ra3/(N u20P r)]1/5
)
, withθI = 2 and θIII= 1. (2.25) For Prandtl numbers small or large enough, (2.25) becomes:
N u0∼
( (Ra√
P r)2/(5θi−2), forδT ≫δu, (2.26a) Ra3/(5θi+2)P r−1/(5θi+2), forδu≫δT. (2.26b) 2.3.3. Grossmann & Lohse (2001) theory for the whole parameter (Ra, P r)plane.
The 4 previous regimes can only be observed experimentally and numerically for extreme values of Ra and P r numbers. For instance regime IV corresponds to very high Ra numbers but in this case ultimate convection could appear while regime II is valid only for very small Ra numbers for which convection is not really turbulent.
Grossmann & Lohse (2001) proposed to describe convection at anyRaandP r numbers as a mixture of these 4 regimes. By replacing the expressions ofhǫuiBu(2.15) andhǫuiBL (2.17) in the balance equation for the viscous dissipation rate (2.10), they obtained this first generalised equation:
RaN u0
P r2 =c1Re5/20
2a +c2Re30. (2.27)
Using (2.19) and (2.21), the second generalised equation can be written as:
N u0=c3
s
Re0P rf
2aN u0
√Re0
+c4Re0P rf
2aN u0
√Re0
. (2.28)
Equations (2.27) and (2.28) give the dependency in Ra and P r of both Re0 and N u0
numbers, assuming the 5 coefficients (a,c1-c4) are known. Stevenset al.(2013) determined these coefficients from previous experimental measurements in the literature.
3. N u vs Ra scalings for internal source driven convection
Using the assumptions discussed below, theN u vs Rascalings presented in previous section for standard RB experiments are generalized for the modified experiments de- scribed in the introduction and in figure 1. The basic assumption is to state that, for high Ranumbers, the dynamical structure of the convective flow is the same in the standard and modified RB experiments. At a constant Ra number, heating in volume produces the same type of thermal boundary layers as those observed in a standard RB cell. The increase in the power of the heating and cooling sources results in an increase in the bulk flow temperature, but the two types of convection experiments are so similar and the mechanisms that control the convective flow are so robust that for both classical and
ultimate regimes, the values ofRa∗ andP e∗are identical in both types of experiments.
For the GL theory, theaparameter and the 4 dimensionless prefactors (1 by regime) are assumed to be independent of the experiment under consideration.
Secondly, in steady state, the equation of heat averaged over a horizontal section can be written as
d(w′T′)
dz −λd2T
dz2 =qv(z). (3.1)
The internal heating and cooling sources are balanced either by convective flux in the bulk flow or by a conductive flux in both boundary layers. Hereafter, only the lower boundary layer will be considered since the upper boundary layer has the same behaviour. In the boundary layer, by neglecting the convective term and using the expression ofqv(z) (see 1.1), (3.1) can be integrated twice to obtain:
Th−T(z) =Qh λ
z h− l
h[1−exp(−z/l)]
. (3.2)
Forz=δT and using the definition of the Nusselt number (1.2), (3.2) yields to 1
2N u =δT
h − l
h[1−exp(−δT/l)]. (3.3) 3.1. Extension of the classical regime given by Malkus
In the classical regime by Malkus, (2.2) yields to δT
h = 2Ra∗
Ra 1/3
= 1
2N u0
. (3.4)
Using (3.4), (3.3) becomes N u
N u0 = 1
1−2 ˜l N u0
h1−exp
−2 ˜l N u1 0i. (3.5) In (3.4) and (3.5), N u0 is the Nusselt number for a standard RB experiment in the classical regime but it also represents the limit ofN u when ˜l = l/h → 0. Even if N u depends on both parameters ˜l andRa, Eq. (3.5) shows that the Nusselt ratioN u/N u0
is a function of a single variable that is the product of ˜landN u0. This is the main result of the present theory and is tested against experimental results in section 4.
The limits of (3.5) when ˜l →0 and ˜lN u0 ≫1 are given in Table 1. It can be noted that, when the product of ˜l andN u0 increases from 0 to ∞, theRa-dependent Nusselt number (N u) increases from a power law of one third to a two thirds, i.e. with an exponent greater than 1/2 which characterizes the ultimate regime for a standard RB experiment (Eq. 2.7).
3.2. Extension of the Kraichnan’s ultimate regime
Unlike the classical regime for which the thickness of the boundary layers depends only onRawhatever the type of experiment considered (see (3.4)), Eq. (2.4) shows that, in the ultimate regime,δT depends on the velocity fluctuations in the bulk (v∗) and therefore on the thermal power injected into the bulk flow. Assuming as before thatv∗∼U/lnRe (Kraichnan 1962), (2.4) becomes for a modified RB experiment
δT
h =P e∗ P r
ln(Re)
Re = (δT)0
h
Re0
ln(Re0) ln(Re)
Re . (3.6)
For a standard RB experiment, (δT)0is given by (2.1) and thus (3.6) becomes δT
h = 1 2N u0
Re0
Re
1 +ln(Re/Re0) ln(Re0)
. (3.7)
As assumed previously for standard RB experiments, the Richardson number in the bulk flow is taken of order 1 i.e. Ri=gα(w′T′)h/U3 =gακQh/(λU3)∼1 yielding to Re3∼RaN u/P r2, similarly to (2.5). Therefore, at constant Rayleigh number, the ratio of the Reynolds numbers for standard and modified RB experiments is proportional to the one-third power law of the ratio of the Nusselt numbers
Re Re0
= N u
N u0
1/3
. (3.8)
Furthermore, (3.8) is valid both for ultimate and classical regimes of convection. Using (3.7) and (3.8), (3.3) can be written as
N2= 1
1 +α−2˜lN u0Nh
1−exp
−2˜lN u1+α0Ni, (3.9) whereN = (N u/N u0)1/3 andα= lnN/lnRe0.
In the ultimate regime and similarly to the classical regime case, the ratio N u/N u0
is a function of the product ˜l×N u0. However,αalso depends on the Rayleigh number through the Reynolds numberRe0. When ˜l→0,α≈0 since on the one handN →1 and on the other Reynolds numbersRe0must be large enough to reach the ultimate regime.
The limit of (3.9) when ˜l →0 is then given in Table 1. For large values of ˜l, (3.9) can be solved numerically for each chosen couple (Ra,˜l) to obtainN and thenN u. At high N u0 or else at very high Rayleigh numbers,N u scales asymptotically as Ra2 i.e. with an exponent 2 well above 1/2 (see Table 1).
3.3. Extension of the GL theory
The balances of the turbulent kinetic energy and of the thermal variance give the following two exact relations (Shraiman & Siggia 1990; Grossmann & Lohse 2000):
hǫui=gα h
"
Z h 0
Φ(z)
ρcp dz−λ∆T ρcp
#
, (3.10)
hǫTi= 1 h
Z h 0
T(z)qv(z) ρcp
dz+ThΦ(0)−TcΦ(h)
ρcph . (3.11)
Actually, for a standard RB experiment, the convective flow is driven by the thermal boundary conditions (∆T orΦ(z= 0)) whereas for the modified RB experiment presented in Fig. 1, the volumetric power source controls the intensity of the convective flow.
Besides, for the second case, the lower and upper plates are assumed to be perfectly insulated conducting toΦ(0) =Φ(h) = 0. In steady state, energy conservation yields to the following relation between the heat flux and volumetric power source:
Φ(z)
Q =
1−exp
−z l
forz6h/2, 1−exp
h−z l
forh/26z6h.
(3.12)
Using (3.12) and (1.2), and assumingN u≫1, (3.10) becomes:
hǫui= ν3 h4
N uRa
P r2 (1− C). (3.13)
The corrective term C = 2˜lh
1−exp
−2˜1li
only depends on ˜l and varies as 2˜l when
˜l → 0. Thus, the expression giving the dissipation rate of kinetic energy averaged over the whole cell (3.13) is very similar to that obtained for a standard RB experiment (2.8).
As for the thermal dissipation rate, its average over the cell is related to the profile of the mean temperature (Eq. 3.11). As the GL theory is based on Prandtl-Blasius- Pohlhausen laminar boundary layers (Grossmann & Lohse 2000), the mean temperature can be written as:
2T(z)−Tb
∆T =
1−ΘP
z δT
forz6h/2, ΘP
h−z δT
−1 forh/26z6h,
(3.14)
withΘP the Pohlhausen temperature profile which is assumed to be independent of the Prandtl number. In particular,ΘP(0) = 0 andΘP(η)→1 whenη≫1. Using (3.14) and (1.1), (3.11) then becomes:
hǫTi=κ ∆T
h 2
N uδT
l
Z h/(2δT) 0
[1−ΘP(η)] exp
−δT
l η
dη. (3.15) Equation (3.15) shows that hǫTi depends both on ˜l=l/h and δT/h. The hypothesis adopted in sub-section 3.1 for extending the classical regime of Malkus is again adopted here (Eq. 3.4). δT/h is assumed to be only controlled by the Rayleigh number so that δT/h= 1/[2N u0(Ra)], whereN u0is the Nusselt number for a standard RB experiment.
Equation (3.15) becomes:
hǫTi=κ ∆T
h 2
N uG(2N u0˜l), (3.16)
with
G(y) =1 y
Z ∞
0
[1−ΘP(η)] exp
−η y
dη. (3.17)
Then, the central idea of the GL theory is to split the dissipation rates into two contributions (see (2.10)-(2.13)). Generalisation of (2.15)-(2.17) are:
hǫuiBu∼ U2 h/U
1−δu
h
≈ν3
h4Re3, (3.18)
hǫTiBu∼ (∆T)2 h/Uedge
1−δT
h
≈κ ∆T
h 2
ReP rf
2aN u0
√Re
, (3.19)
hǫuiBL∼ν U
δu
2
δu
h = ν3 h4
Re5/2
2a . (3.20)
To obtain (3.19), the relevant velocity at the edge between the thermal BL and bulk is assumed to be expressed as Uedge =U f(δu/δT), with the same function f used for standard RB convection. Besides, we have
δu
δT
= 2N u0
√Re. (3.21)
Combining (3.13) and (3.18), and using (2.5), we obtain the following first equation valid for both regimesII andIV:
N u N u0
(1− C) = Re
Re0
3
(regimesII and IV). (3.22) Forgetting the corrective term C which must be small since ˜l ≪ 1, (3.22) is the same equation as the one obtained both for extending the classical and ultimate regimes of Malkus and Kraichnan (see Eq. 3.8). On the contrary, for regimesIandIII, (3.22) needs to be replaced by:
N u N u0
(1− C) = Re
Re0
5/2
(regimesI andIII). (3.23) The results of this theory is first presented for regime IV because most of the experimental and numerical results fall into this regime. In addition, unlike regimeII, the extension of regimeIV to internally heated convection does not require the introduction of any adjustable parameters.
3.3.1. Regime IV,hǫui ∼ hǫuiBu andhǫTi ∼ hǫTiBu
For regime IV, the thermal dissipation rate is dominated by its bulk contribution.
Combining (3.16) and (3.19), and using (2.19), we obtain:
N u
N u0G(2N u0˜l) = Re Re0
f(2aN u0/√ Re) f(2aN u0/√
Re0). (3.24)
The system of equations (3.22) and (3.24) gives the dependency of both N u/N u0 and Re/Re0 as a function of the 3 control parameters: Ra, P r and ˜l = l/h. For Prandtl numbers small or large enough, (3.24) can be simplified as follows:
N u N u0
=
( (1− C)−12[G(2N u0˜l)]−32, forδT ≫δu (regimeIVl) (3.25a) (1− C) [G(2N u0˜l)]−2, forδu≫δT (regime IVu). (3.25b) Besides, the limits of (3.25a) and (3.25b) when ˜lN u0 tends to 0 or ∞can be obtained saying that G(y) y→0≈ 1−Θ′P(0)y or G(y)y→∞≈ δdΘ/y, where δΘd =R∞
0 [1−θP(η)]dη. A summary of the corresponding results is given in Table 1.
3.3.2. Regime II, hǫui ∼ hǫuiBu andhǫTi ∼ hǫTiBL
Following the idea of Grossmann & Lohse (2000), we consider the order of magnitude of the different terms of energy equation i.e.ux∂xT+uz∂zT =κ∂zzT+ρcqvp. It yields to
Uedge∆T
h ∼κ∆T
δ2T +Aqv(δT) ρcp
. (3.26)
Using (1.1), (1.2), (2.1), (2.21) andUedge=U f(δu/δT), (3.26) becomes:
1 + ˜A N u
N u0H(2N u0˜l) = Re Re0
f(2aN u0/√ Re) f(2aN u0/√
Re0), (3.27)
with H(y) = 1yexp
−y1
, 0 6H(y)6exp(−1) ≈0.37, and ˜A a numerical constant of the order of one to be determined experimentally.
N u0 Nu
Nu0−1 for ˜l N u0≪1 N u for ˜l N u0≫1 Classical regime
by Malkus, Eq. (3.5) ∼Ra13 2 ˜l N u0
∼˜l Ra13
4 ˜l N u20
∼ ˜l Ra23 Ultimate regime
by Kraichnan, Eq. (3.9) ∼(CU0Ra)12 3 ˜l N u0
∼˜l(CU0Ra)12
(1+2α)6 ˜l3N u40
∼ ˜l3(C0URa)2 RegimeIVl
by GL, Eq. (3.25a) ∼(RaP r)12 3ΘP′ (0) ˜l N u0
∼˜l(RaP r)12
2
δdΘ
3/2
˜l32N u
5 2 0
∼˜l32(RaP r)54 RegimeIVu
by GL, Eq. (3.25b) ∼Ra13 4ΘP′ (0) ˜l N u0
∼˜l Ra13
2
δdΘ
2
˜l2N u30
∼ ˜l2Ra RegimeIIIu
by GL, Eq. (3.30) ∼Ra37P r−17 5ΘP′ (0) ˜l N u0
∼˜l Ra37P r−17
2
δdΘ
52
˜l52N u
7 2 0
∼ ˜l52 Ra32P r−12 Table 1.Limits when ˜l N u0≪1 and ˜l N u0≫1 of the Nusselt number for a radiatively heated convection experiment. For the ultimate regime by Kraichnan,α= ln(N u/N u0)/(3 lnRe0) and C0U=P r/(P e∗lnRe0)3. For regimesIV andIII by GL, we assume that ˜l≪1 to haveC →0.
Equations (3.22) and (3.27) give the dependency of bothN u/N u0 and Re/Re0 as a function ofRa,P r and ˜l=l/h. Contrary to the regimeIV,Re/Re0andN u/N u0 both tend towards 1 whenN u0˜l≫1 for regimeII because H(y)≈1/ywheny≫1. For the two limitsN u0˜l≪1 andN u0˜l≫1, we obtain:
N u N u0
(1− C) = 1 + A θ˜ i
1− CH(2N u0˜l), (3.28) withθi= 3 forδT ≫δu (regimeIIl) andθi= 2 forδu≫δT (regimeIIu).
For any value ofN u0˜l, we show in appendix A thatN u/N u0can be given with a very good approximation by:
N u
N u0(1− C) =
"
Sβ0 A˜H 1− C
!#3
, (3.29)
with Sβ(x) the real and positive solution of the equation: 1 +xSβ3 =Sβ1+β/2, andβ0= h2aN u0
√Re0 f(2aN u√Re0
0 )i−n
.
3.3.3. Regime I,hǫui ∼ hǫuiBL andhǫTi ∼ hǫTiBL
For regimeI, (3.23) and (3.27) giveN u/N u0andRe/Re0as a function ofRa,P rand
˜l =l/h. For the two limits N u0˜l ≪ 1 andN u0˜l ≫1, (3.28) is valid with θi = 5/2 for δT ≫δu (regimeIl) andθi= 5/3 forδu≫δT (regimeIu).
10−2 100 102 1
10 50
106 108 1010 1012 1
10 50
10−4 10−3 10−2 10−1 1
10 50
10−1 100
1 10 50
˜l
Ra P r
˜l N u0
N u N u0
N u N u0
N u N u0
N u N u0
(a) (b)
(c) (d)
Figure 2.Results of the extension of GL theory for radiatively heated convection using (3.33) and (3.34) with ˜A= 0.3. The coefficientaand prefactorsc1-c4are given by Stevenset al.(2013).
(a)N u/N u0 versusRa forP r= 0.01 (red dotted lines) and P r= 1 (blue dashed lines), and for ˜l= 10−3(lowest line), ˜l= 10−2(middle line) and ˜l= 0.1 (highest line). (b)N u/N u0 versus P rforRa= 108 (red dotted lines) andRa= 1010 (blue dashed lines), and for ˜l= 0.01 (lowest line), ˜l= 0.02 (middle line) and ˜l= 0.05 (highest line). (c)N u/N u0versus ˜lforP r= 1 and for Ra= 106(lowest line), 108, 1010, 1012and 1014(highest line). (d) Same results as (c) but using
˜l×N u0 asx-coordinate. Also shown: regimeIVu(N u/N u0 =G−2) (blue solid line), regimeIVl
(N u/N u0 =G−3/2) (blue dashed line), regimeIIu(red solid line) and regimeIIl (red dashed line).
3.3.4. Regime III,hǫui ∼ hǫuiBLandhǫTi ∼ hǫTiBu
For regimeIII, (3.23) and (3.24) giveN u/N u0and Re/Re0 as a function ofRa,P r and ˜l=l/h. For Prandtl numbers large enough, we obtain:
N u N u0
= (1− Cu)3/2[G(2N u0˜l)]−5/2 forδu≫δT (regimeIIIu). (3.30) The limits of (3.30) when ˜l×N u0tends to 0 or ∞are given in Table 1.
3.3.5. Theory in the whole parameter (Ra, P r,˜l)plane
Following the idea of Grossmann & Lohse (2001) (see paragraph 2.3.3), at givenRa, P rand ˜l, radiatively driven convection can be described as a mixture of these 4 regimes.
By replacing the expressions ofhǫuiBu(3.18) andhǫuiBL(3.20) in the balance equation
for the viscous dissipation rate (3.13), the first generalised equation can be written as:
RaN u
P r2 (1− C) =c1
Re5/2
2a +c2Re3. (3.31)
Using (3.24) and (3.29), the second generalised equation becomes:
N u= c3
1− C
"
Sβ A˜H 1− C
!#3s
Re0P rf
2aN u0
√Re0
+c4
GReP rf
2aN u0
√Re
. (3.32) By combining on the one hand (2.27) and (3.31), and on the other hand (2.28) and (3.32), we obtain the two equations which giveN uandRe numbers as a function of the 3 parametersRa,P r and ˜l:
N u N u0
= 1
1− C c1Re5/2
2a +c2Re3 c1Re5/20
2a +c2Re30
, (3.33)
N u N u0 =
c3
1−C
hSβ ˜
AH 1−C
i3r
Re0P rf
2aN u√ 0
Re0
+cG4ReP rf
2aN u√ 0
Re
c3
r
Re0P rf
2aN u0
√Re0
+c4Re0P rf
2aN u0
√Re0
. (3.34)
Figures 2 (a) and (b) show the variations of the ratio N u/N u0 againstRa andP r for fixed values of ˜l, whileN u/N u0is plotted against ˜l in Fig. 2 (c) forP r= 1 and for fixed values ofRabetween 106and 1014. As observed previously for the extensions of Malkus and Kraichnan theories, the use of the variableN u0טlallows to gather the various curves drawn in Fig. 2 (c) (see Fig. 2d). As underlined by Grossmann & Lohse (2001) for RB convection, pure regime IVu is only reached for very high Ra and P r numbers (blue upper solid line in Fig. 2d) while for moderate values ofRaandP rnumbers, radiatively heated convection is described by a mixing of the 4 regimesI-IV.
4. Comparison with experimental results
The predictions of this theoretical approach can be tested thanks to the recent exper- imental investigation of Lepot et al.(2018); Bouillautet al.(2019). The measurements cover a range of 4×106 to 4×109 forRa, 5×10−5 to 0.1 for ˜l =l/hand the working fluid was water so the Prandtl number was set at a constant value close to 7. Here, the Rayleigh and Nusselt numbers are defined using ∆T = 2(Th−Tb), where Th and Tb
are the measured temperature of the lower plate and the bulk flow, respectively. Hence, there is a factor 2 forRa(and a factor 1/2 forN u) by comparing the figures from Lepot et al.(2018); Bouillaut et al.(2019) and with those presented here. Instead of plotting N uas a function ofRa, the theory presented in section 3 shows that Nusselt numbers for various ˜lshould better collapse around a single curve by plotting the ratio of the Nusselt numbers for modified and standard RB experiments (N u/N u0) against the product of ˜l andN u0. As underlined by Lepotet al.(2018); Bouillautet al.(2019), their experiments converge to RB experiments when ˜l→0, even though the boundary conditions are very different (the horizontal plates are insulated while they are perfectly conductive for RB convection). Indeed, for ˜l = 5×10−5, Nusselt numbers given by Bouillautet al.(2019) can be fitted by a simple scaling such asN u= 0.076×Ra1/3with a maximum deviation of 7%, or by the GL theory witha= 0.75,c1= 8.05,c2 = 1.38,c3= 0.3 andc4= 0.03.
Prefactorsc1andc2are the ones given by Stevenset al.(2013) whilec3andc4are slightly modified to better fit to the experimental results (Stevenset al.(2013) advocated taking
10−1 100 101 1
10 50
˜l N u0
N u N u0
Figure 3. Compensated Nusselt numbers for a radiatively heated convection experiment as a function of ˜l N u0. Coloured symbols: experiments from Bouillautet al.(2019) with ˜l= 0.0015 (⋆), ˜l= 0.003 (+), ˜l= 0.006 (♦), ˜l = 0.0012 (∗), ˜l= 0.0024 (), ˜l = 0.048 (⊲), ˜l = 0.05 (◦),
˜l= 0.096 (∇). Black dashed line: extension of the classical regime by Malkus (1954), Eq. (3.5), no adjustable parameter. Blue solid line: extension of the classical regimeIVuby Grossmann &
Lohse (2000), Eq. (3.25b) withC= 0, no adjustable parameter. Grey symbols: Eqs. (3.33) and (3.34) with ˜A= 0.35. Upper red lines: extension of the ultimate regime by Kraichnan (1962), Eq. (3.9) withRe0= 1000 (red dotted line) andRe0= 1010 (red dashed line).
c3= 0.487 andc4= 0.0252). Using GL theory for definingN u0, the experimental results of Bouillautet al.(2019) are plotted in Fig. 3 for 0.00156˜l60.01 (coloured symbols).
The black dashed and blue solid lines represent, respectively, the extension of the classical scaling proposed by Malkus (1954) (Eq. 3.5) and the extension of regime IVu proposed by Grossmann & Lohse (2000) (Eq. 3.25b). First, instead of plotting N u against Ra (see Fig. 2 in Bouillaut et al.(2019)), plotting compensated Nusselt numbersN u/N u0
as a function of ˜l N u0 ∼ ˜l Ra1/3 allows to collapse the experimental data on a single curve. Secondly, this curve is given by (3.5) or (3.25b) with a fairly good accuracy and without the use of any adjustable parameter. The system of equations (3.33) and (3.34) that results from a mixture of regimes I to IV and is represented by grey symbols in Fig. 3 gives slightly lower values for the ratioN u/N u0 than the pure regimeIV (blue solid line). The new parameter ˜A has little impact on the curve for this data set and is fixed to 0.35 in Fig. 3. In view of: (i) the experimental uncertainties, (ii) the product
˜l×N u0 is always less than 5 and (iii) P r number is fixed to 7 for the experimental results, it is difficult to discriminate between the different extensions of the theoretical models presented in section 3 describing convection in classical regimes.
On the contrary, it is well known that convection in the so-called ultimate regime behaves very differently sinceN u0scales asymptotically asRa1/2, thus with an exponent 1/2 much higher than 1/3. The theoretical work presented in section 3 shows that, for radiatively heated convection, there is also a clear difference for the ratioN u/N u0
between classical regimes and the ultimate regime. In figure 3, the two upper red lines represent the ultimate regime (Eq. 3.9) for two fixed Reynolds numbers (dotted line:
Re0 = 1000, dashed line: Re0 = 1010). They are clearly above all other curves and symbols describing the theoretical and experimental results for classical regimes. Indeed, Ra numbers achieved by the experiments of Bouillautet al. (2019) are not sufficient to trigger the ultimate regime (a detailed discussion is given in appendix B).
5. Conclusions
The well known theories of RB convection have been extended here to radiatively heated convection. The evolution of the Nusselt number as a function of Ra, P r and l/h (where l is the heating length near the lower plate and hthe height of the cell) is predicted whatever the convection regime considered. In the classical regime and using the simple theory of Malkus (1954), equation (3.5) gives N u/N u0 as a function of Ra andl/hwithout adjustable parameter, while, considering the more recent Grossmann &
Lohse (2000) theory, the two equations (3.33) and (3.34) give the dependency of both N u/N u0 and Re/Re0 as a function of the 3 control parameters Ra, P r and l/h. It can be noted that only the extension of regimesI and II of the GL theory, observable only at lowRa numbers, needs an adjustable parameter. A good agreement is observed between the experimental results obtained by Bouillautet al.(2019) and the theoretical results for the classical regime. For the ultimate regime, equation (3.9) gives the Nusselt number as a function of Ra and l/h without adjustable parameter, but, in this case, no experimental or numerical results exist to test this prediction. Finally, this work predicts that the Nusselt number behaves asymptotically asRa2/3orRafor the classical regime (see Table 1) while it scales asRa2in the ultimate regime, and this prediction is of major interest for geophysical and astrophysical flows where convection is driven by internal heat sources.
Acknowledgements
Bernard Castaing is gratefully thanked for his suggestions and the review of the article.
Declaration of interests
The author reports no conflict of interest.
Appendix A. Extension of regime II of the GL theory
For regime II, Eqs. (3.22) and (3.27) give the dependency of bothN u/N u0andRe/Re0
as a function ofRa,P r and ˜l=l/h. Using (3.22), (3.27) becomes:
1 + A˜H 1− C
Re Re0
3
= Re Re0
f
2aN u0
√Re
f
2aN u0
√Re0
. (A 1)
A.1. RegimeIIu (highP r numbers or δu≫δT) Equation (A 1) becomes:
1 + A˜H 1− C
Re Re0
3
= Re
Re0
3/2
. (A 2)
To get a positive value forRe, we must have: ˜AH6(1− C)/461/4. AsH6exp(−1), the parameter ˜Aneeds to be lower than exp(1)/4≈0.68. Resolution of (A 2) gives:
Re Re0
=Su A˜H 1− C
!
, (A 3)
ts
10−1 100 101
1 1.5
2 2.5 3 3.5
10−1 100 101
1 1.2 1.4 1.6 1.8 2
˜l N u0 ˜l N u0
N u N u0
N u N u0
(a) (b)
Figure 4. N u/N u0 versus ˜l N u0 in the regime II of the extension of GL theory for radiatively heated convection. Grey symbols: solution of Eqs. (3.22) and (3.27) with C = 0, A˜= 4 exp(1)/27≈0.40 (a) and ˜A= 0.35 (b). Black solid lines: Eq. (3.29). From top to bottom, the parameter 2aN u0/√
Re0is taken equal to 0.5, 1 and 1.5. The lower solid red line represents regimeIIuwhile the upper dashed red line shows regimeIIl.
with Su(x) =
1−√ 1−4x 2x
2/3
. Su(x) is an increasing function of x withSu(0) = 1 andSu(1/4) = 22/3≈1.59, yielding to 16Re/Re061.59.
A.2. RegimeIIl (low P r numbers orδT ≫δu) Equation (A 1) becomes:
1 + A˜H 1− C
Re Re0
3
= Re Re0
. (A 4)
To get a positive value forRe, we must have: ˜AH64(1−C)/2764/27. AsH6exp(−1), the parameter ˜Aneeds to be lower than 4 exp(1)/27≈0.40. Resolution of (A 4) gives:
Re Re0
=Sl A˜H 1− C
!
, (A 5)
withSl(x) = 2
√3xcos
"
1
3arccos 3√ 3x 2
! +π
3
#
.Sl(x) is an increasing function ofxwith Sl(0) = 1 andSl(4/27) = 3/2 yielding to 16Re/Re063/2.
A.3. Approximation of (A 1) for anyP r numbers
For the two limits P r ≫ 1 and P r ≪ 1, we have shown that 16Re/Re063/2.
The following approximation can then be adopted: f(x0) ≈ f(x)(xx0)−β0, with β0 =
−d lnf
d lnx
x=x0
= [x0f(x0)]−n (n = 4). When x0 ≫ 1, β0 → 3/2 while β0 → 0 when x0→0. Equation (A 1) can therefore be approximated by:
1 + A˜H 1− C
Re Re0
3
= Re
Re0
1+β0/2
. (A 6)
